\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 49, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/49\hfil Existence and uniqueness theorems]
{Existence and uniqueness theorems on certain
difference-differential equations}

\author[B. G. Pachpatte\hfil EJDE-2009/49\hfilneg]
{Baburao G. Pachpatte} 

\address{Baburao G. Pachpatte \newline
 57 Shri Niketan Colony, Near Abhinay Talkies, Aurangabad 431 001
(Maharashtra), India}
\email{bgpachpatte@gmail.com}

\thanks{Submitted June 9, 2008. Published April 7, 2009.}
\subjclass[2000]{34K5, 34K10, 34K40}
\keywords{Existence and uniqueness; difference-differential equations;
\hfill\break\indent
Leray-Schauder alternative;  Bihari's integral inequality;
Wintner type; Osgood type}

\begin{abstract}
 In the present paper, existence and uniqueness theorems for the
 solutions of certain nonlinear difference-differential equations
 are established. The main tools employed in the analysis are
 based on the applications of the Leray-Schauder alternative and
 the well known Bihari's integral inequality.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

Let $ \mathbb{R}^n $ denote the real $n$-dimensional Euclidean
space with appropriate norm denoted by $ |\cdot | $ and
$J = [{0,T}]$ ($T > 0$), $\mathbb{R}_+  = [0,\infty) $ be the given
subsets of $\mathbb{R}$, the set of real
numbers. In this paper, we consider the difference-differential
equation of the form
\begin{equation}
x'(t) = f({t,x(t),x({t - 1})}), \label{e1.1}
\end{equation}
for $t \in J $ under the initial conditions
\begin{equation}
x({t - 1}) = \phi (t)\;({0 \le t< 1}),\quad x(0) = x_0 , \label{e1.2}
\end{equation}
where $ f \in C({J \times \mathbb{R}^n  \times \mathbb{R}^n ,\mathbb{R}^n})$
 and $ \phi (t) $ is a continuous function for $ 0 \le t < 1$,
$\lim _ {t \to 1 - 0} \phi (t)$ exists, for which we denote by
$\phi ({1 - 0}) = c_0 $.
 If we consider the solutions of  \eqref{e1.1} for ${t \in J}$,
 we obtain a function $x(t-1)$ which is unable to define as a solution for
$0 \le t < 1$. Hence, we have to impose some condition, for example
the first condition in \eqref{e1.2}. We note that, if $T$ is less than $1$,
the problem is reduced to ordinary differential equation
\begin{equation}
x'(t) = f({t,x(t),\phi (t)}), \label{e1.3}
\end{equation}
for $0 \le t < 1$, with the second condition in \eqref{e1.2}. Here,
it is essential to obtain the solutions of equation \eqref{e1.1} for
$0 \le t \le T$, so that, we suppose, in the sequel, $T$ is not less than $1$. 
It is easy to observe that the integral equations which are equivalent 
to \eqref{e1.1}-\eqref{e1.2} are
\begin{equation}
x(t) = x_0  + \int_0^t {f({s,x(s),\phi (s)})} ds,\label{e1.4}
\end{equation}
for $0 \le t < 1$ and
\begin{equation}
x(t) = x_0  + \int_0^1 {f({s,x(s),\phi (s)})} ds 
+ \int_1^t {f({s,x(s),x({s - 1})})} ds,
\label{e1.5}
\end{equation}
for $1 \le t \le T$.

Problems of existence and uniqueness of solutions of equations of
the form \eqref{e1.1} and its more general versions, under various
initial conditions have been studied by many authors by using
different techniques. The fundamental tools used in the existence
proofs, are essentially, the method of successive approximation,
Schauder-Tychonoff's fixed point theorem, Banach contraction
mapping principle and comparison method, see [2,6,9,12,14,15] and
the references cited therein. Our main objective here is to
investigate the global existence of solution to
\eqref{e1.1}-\eqref{e1.2} by using simple and classical
application of the topological transversality theorem of Granas
[5, p. 61], also known as Leray-Schauder alternative. Osgood type
uniqueness result for the solutions of \eqref{e1.1}-\eqref{e1.2}
is established by using the well known Bihari's integral
inequality. Existence and uniqueness theorems for certain
perturbed difference-differential equation are also given.

\section{Main Results}

In proving existence of solution of \eqref{e1.1}-\eqref{e1.2}, we
use the following topological transversality theorem given by
Granas [5, p.61].

\begin{lemma} \label{lem1}
Let $B$ be a convex subset of a normed linear space $E$ and assume
$0 \in B$. Let $ F:B \to B $ be a completely continuous operator
and let $ U(F) = \{ {x:x = \lambda Fx}\} $ for some $ 0 < \lambda
< 1$. Then either $U(F)$ is unbounded or $F$ has a fixed point.
\end{lemma}

We also need the following integral inequality, often referred to
as Bihari's inequality [8, p. 107].

\begin{lemma} \label{lem2}
Let $u(t),p(t) \in C({\mathbb{R}_+  ,\mathbb{R}_+  })$. Let
$w(u)$ be a continuous, nondecreasing function defined on
${\mathbb{R}_+ } $, $w(u)>0$ for $u>0$ and $w(0)=0$. If
\[
u(t) \le c + \int_0^t {p(s)w({u(s)})ds,}
\]
for $t \in \mathbb{R}_+$,  where $ c \ge 0 $ is a constant, then
for $ 0 \le t \le t_1$,
\[
u(t) \le W^{ - 1} \Big[{W(c) + \int_0^t {p(s)ds} }\Big],
\]
where
\[
W(r) = \int_{r_0 }^r {\frac{{ds}}{{w(s)}}} ,\quad r > 0,\; r_0>0,
\]
$W^{ - 1}$ is the inverse function of $W$ and ${t_1  \in \mathbb{R}_+  }$
be chosen so that
\[
{W(c) + \int_0^t {p(s)ds \in \mathop{\rm Dom}({W^{ - 1} }),} }
\]
for all ${t \in \mathbb{R}_+  }$ lying in the interval $0 \le t \le t_1$.
\end{lemma}

The following theorem deals with the Wintner type global existence result
for the solution of \eqref{e1.1}-\eqref{e1.2}.

\begin{theorem} \label{thm1}
Suppose that the function $f$ in \eqref{e1.1} satisfies the condition
\begin{equation}
| {f({t,x,y})} | \le h(t)[{g({| x |}) + g({| y |})}],
\label{e2.1}
\end{equation}
where $h \in C({J,R})$ and $g:\mathbb{R}_+   \to ({0,\infty })$ is
continuous and nondecreasing function.
 Then \eqref{e1.1}-\eqref{e1.2} has a solution $x(t)$ defined on $J$
provided $T$ satisfies
\begin{equation}
\int_0^T {[{h(s) + h({s + 1})}]ds < \int_\alpha ^\infty  {\frac{{ds}}{{g(s)}},} }
\label{e2.2}
\end{equation}
where
\begin{equation}
\alpha  = | {x_0 } | + \int_0^1 {h(s)} g({| {\phi (s)} |})ds.
\label{e2.3}
\end{equation}
\end{theorem}

\begin{proof} The proof will be given in three steps.

\noindent\textbf{Step I.} To  use Lemma 1, we establish the priori
bounds on the solutions of the problem
\begin{equation}
x'(t) = \lambda f({t,x(t),x({t - 1})}), \label{e2.4}
\end{equation}
under the initial conditions \eqref{e1.2} for $ \lambda  \in ({0,1})$.
Let $x(t)$ be a solution of \eqref{e2.4}-\eqref{e1.2}, then we
consider the following two cases.

Case 1: $ 0 \le t < 1$. From the hypotheses, we have
\begin{equation}
\begin{aligned}
| {x(t)} | &= \big| {x_0  + \int_0^t {\lambda f({s,x(s),\phi (s)})ds} }
\big| \\
& \le | {x_0 } | + \int_0^t {h(s)[{g({| {x(s)} |})
+ g({| {\phi (s)} |})}]ds} \\
& = \alpha  + \int_0^t {h(s)g({| {x(s)} |})ds.}
\end{aligned}\label{e2.5}
\end{equation}
Let $u(t)$ be defined by the right hand side of \eqref{e2.5}, then
$u(0) = \alpha ,| {x(t)} | \le u(t)$ and
\[
u'(t) = h(t)g({| {x(t)} |}) \le h(t)g({u(t)});
\]
that is,
\begin{equation}
\frac{{u'(t)}}{{g({u(t)})}} \le h(t).
\label{e2.6}
\end{equation}
Integration of \eqref{e2.6} from $0$ to $t$ ($0 \le t < 1$),
the change of variable, and the condition \eqref{e2.2} gives
\begin{equation}
\int_\alpha ^{u(t)} {\frac{{ds}}{{g(s)}}}  \le \int_0^t {h(s)ds
\le \int_0^1 {h(s)ds}  < \int_\alpha ^\infty  {\frac{{ds}}{{g(s)}}.} }
\label{e2.7}
\end{equation}
 From this inequality,  we conclude that, there is a constant
$Q_1$ independent of $\lambda  \in ({0,1})$ such that
$u(t) \le Q_1$ for $0 \le t < 1$ and hence ${| {x(t)} | \le Q_1 }$.

Case 2: $ 1 \le t \le T$. From the hypotheses, we have
\begin{equation}
\begin{aligned}
| {x(t)} |
&= | {x_0  + \int_0^1 {\lambda f({s,x(s),\phi (s)})} ds
  + \int_1^t {\lambda f({s,x(s),x({s - 1})})} ds} | \\
& \le | {x_0 } | + \int_0^1 {h(s)[{g({| {x(s)} |})
  + g({| {\phi (s)} |})}]ds} \\
&\quad + \int_1^t {h(s)[{g({| {x(s)} |})
  + g({| {x({s - 1})} |})}]ds} \\
&= | {x_0 } | + \int_0^1 h(s)g({| {\phi (s)} |})ds
  +  \int_0^1 {h(s)g({| {x(s)} |})ds} \\
&\quad  + \int_1^t h(s)g({| {x(s)} |})ds +
  \int_1^t {h(s)g({| {x({s - 1})} |})ds} \\
& = \alpha  + \int_0^t {h(s)g({| {x(s)} |})ds + I_1 }
\end{aligned}\label{e2.8}
\end{equation}
where
\begin{equation}
I_1  = \int_1^t {h(s)g({| {x({s - 1})} |})ds} .\label{e2.9}
\end{equation}
By the change of variable, from \eqref{e2.9}, we observe that
\begin{equation}
I_1  = \int_0^{t - 1} {h({\sigma  + 1})g({| {x(\sigma)} |})d\sigma }
\le \int_0^t {h({\sigma  + 1})g({| {x(\sigma)} |})d\sigma .}
\label{e2.10}
\end{equation}
Using this inequality in \eqref{e2.8}, we obtain
\begin{equation}
| {x(t)} | \le \alpha  + \int_0^t {[{h(s) + h({s + 1})}]g({| {x(s)} |})ds.}
\label{e2.11}
\end{equation}
Let $v(t)$ be defined by the right hand side of \eqref{e2.11}, then
$v(0) = \alpha$, $| {x(t)} | \le v(t)$ and
\[
v'(t) = [{h(t) + h({t + 1})}]g({| {x(t)} |})
\le [{h(t) + h({t + 1})}]g({v(t)});
\]
that is,
\begin{equation}
\frac{{v'(t)}}{{g({v(t)})}} \le [{h(t) + h({t + 1})}].
\label{e2.12}
\end{equation}
Integration of \eqref{e2.12} from $0$ to $t$, $1 \le t \le T$,
the change of variable, and the condition \eqref{e2.2} give
\begin{equation}
\int_\alpha ^{v(t)} {\frac{{ds}}{{g(s)}}}  \le \int_0^t {[{h(s)
+ h({s + 1})}]ds \le \int_0^T {[{h(s) + h({s + 1})}]ds}  <
\int_\alpha ^\infty  {\frac{{ds}}{{g(s)}}.} } \label{e2.13}
\end{equation}
 From \eqref{e2.13} we conclude that there is a constant $ Q_2 $
independent of $ \lambda  \in ({0,1}) $ such that $ v(t) \le Q_2$
and hence $ | {x(t)} | \le Q_2 $ for $ 1 \le t \le T$.
Let $ Q = \max \{ {Q_1 ,Q_2 }\}$. Obviously,
$ | {x(t)} | \le Q $ for $ t \in J $ and consequently,
$\| x\| = \sup \{ {| {x(t)} |:t \in J}\} \le Q$.

\noindent\textbf{Step II.}
We define $ B = C({J,\mathbb{R}^n }) $ to be the Banach space of
all continuous functions from $J$ into $ {\mathbb{R}^n } $
endowed with sup norm defined above. We rewrite the problem
\eqref{e1.1}-\eqref{e1.2} as follows. If $ y \in B $ and
$ x(t) =y(t) + x_0$, it is easy to see that
\[
y(t) = \int_0^t {f({s,y(s) + x_0 ,\phi (s)})ds,}
\]
for $0 \le t < 1$ and
\[
y(t) = \int_0^1 {f({s,y(s) + x_0 ,\phi (s)})ds}
+ \int_1^t {f({s,y(s) + x_0 ,y({s - 1}) + x_0 })ds,}
\]
for $1 \le t \le T$,
$y(0) = y_0  = 0$
if and only if $x(t)$ satisfies \eqref{e1.1}-\eqref{e1.2}.
Let $B_0  = \{ {y \in B:y_0  = 0} \}$ and define
$F:B_0  \to B_0$ by
\begin{equation}
Fy(t) = \int_0^t {f({s,y(s) + x_0 ,\phi (s)})ds,}
\label{e2.14}
\end{equation}
for $0 \le t < 1$ and
\begin{equation}
Fy(t) = \int_0^1 {f({s,y(s) + x_0 ,\phi (s)})ds}  + \int_1^t
{f({s,y(s) + x_0 ,y({s - 1}) + x_0 })ds,} \label{e2.15}
\end{equation}
 for $1 \le t \le T$. Then $F$ is clearly continuous. Now, we shall
prove that $F$ is uniformly bounded. Let $ \{ {b_m }\}$ be a bounded
sequence in $ B_0 $, that is, $\| {b_m }\| \le b $ for all m,
where $b>0$ is a constant. Let $ N =
\sup \{ {h(t):t \in J} \}$. We have to consider the
two cases.

Case 1: $0 \le t < 1$. From \eqref{e2.14} and hypotheses, we have
\begin{equation}
\begin{aligned}
| {Fb_m (t)} |
&\le \int_0^t {| {f({s,b_m (s) + x_0 ,\phi (s)})} |ds} \\
&\le \int_0^t {h(s)[{g({| {b_m (s)} | + | {x_0 } |})
 + g({| {\phi (s)} |})}]ds} \\
&\le \int_0^1 {h(s)g({| {\phi (s)} |})ds}  + \int_0^t {h(s)g({b
+ | {x_0 } |})ds} \\
&\le \gamma  + \int_0^1 Ng({b + | {x_0 } |})ds\\
&= \gamma  + Ng({b + | {x_0 } |}).
\end{aligned} \label{e2.16}
\end{equation}
where
\begin{equation}
\gamma  = \int_0^1 {h(s)g({| {\phi (s)} |})ds} .\label{e2.17}
\end{equation}

Case 2: $1 \le t \le T$. From \eqref{e2.15} and the hypotheses, we have
\begin{equation}
\begin{aligned}
| {Fb_m (t)} | &\le \int_0^1 {| {f({s,b_m (s) + x_0 ,\phi (s)})} |ds}
  + \int_1^t {| {f({s,b_m (s) + x_0 ,b_m ({s - 1}) + x_0 })} |ds}\\
& \le \int_0^1 {h(s)[{g({| {b_m (s)} | + | {x_0 } |})
 + g({| {\phi (s)} |})}]ds} \\
&\quad + \int_1^t {h(s)[{g({| {b_m (s)} | + | {x_0 } |})
 + g({| {b_m ({s - 1})} | + | {x_0 } |})}]ds} \\
& = \int_0^1 {h(s)g({| {\phi (s)} |})ds + \int_0^1 {h(s)g({| {b_m (s)} |
 + | {x_0 } |})ds} } \\
&\quad  + \int_1^t {h(s)g({| {b_m (s)} | + | {x_0 } |})ds}
  + \int_1^t {h(s)g({| {b_m ({s - 1})} | + | {x_0 } |})ds}\\
& = \gamma  + \int_0^t {h(s)g({| {b_m (s)} | + | {x_0 } |})ds + I_2 ,}
\end{aligned} \label{e2.18}
\end{equation}
where $\gamma$ is given by \eqref{e2.17}, and
\begin{equation}
I_2  = \int_1^t {h(s)g({| {b_m ({s - 1})} | + | {x_0 } |})ds} .\label{e2.19}
\end{equation}
By the change of variable, we have
\begin{equation}
I_2  = \int_0^{t - 1} {h({\sigma  + 1})g({| {b_m (\sigma)} |
 + | {x_0 } |})d\sigma }  \le \int_0^t {h({\sigma  + 1})g({| {b_m (\sigma)} |
 + | {x_0 } |})d\sigma .}
\label{e2.20}
\end{equation}
Using \eqref{e2.20} in \eqref{e2.18}, we have
\begin{equation}
\begin{aligned}
| {Fb_m (t)} | &\le \gamma  + \int_0^t {[{h(s) + h({s + 1})}]g({| {b_m (s)} |
+ | {x_0 } |})ds} \\
& \le \gamma  + \int_0^T {2Ng({b + | {x_0 } |})ds}  \\
&= \gamma  + 2NTg({b + | {x_0 } |}).
\end{aligned} \label{e2.21}
\end{equation}
 From \eqref{e2.16} and \eqref{e2.21}, it follows that
$\{ {Fb_m } \}$ is uniformly bounded.

\noindent\textbf{Step III.}
 We shall show that the sequence $\{ {Fb_m } \}$ is equicontinuous.
Let $\{ {b_m } \}$ and $N$ be as in Step II. We must consider three cases.

Case 1: $t$ and ${t'} $ are contained in $ 0 \le t < 1$.
 From \eqref{e2.14}, it follows that
\begin{equation}
\begin{aligned}
Fb_m (t) - Fb_m ({t'})
&= \int_0^t {f({s,b_m (s) + x_0 ,\phi (s)})ds}  - \int_0^{t'} {f({s,b_m (s)
 + x_0 ,\phi (s)})ds} \\
&= \int_{t'}^t {f({s,b_m (s) + x_0 ,\phi (s)})ds} .
\end{aligned}\label{e2.22}
\end{equation}
 From the above equality and hypotheses, we have
\begin{equation}
\begin{aligned}
| {Fb_m (t) - Fb_m ({t'})} |
&\le | {\int_{t'}^t {| {f({s,b_m (s) + x_0 ,\phi (s)})} |ds} } | \\
&\le | {\int_{t'}^t {h(s)[{g({| {b_m (s)} | + | {x_0 } |})
 + g({| {\phi (s)} |})}]ds} } | \\
&\le | {\int_{t'}^t {N[{g({b + | {x_0 } |}) + g({| {c_0 } |})}]ds} } |\\
& = N[{g({b + | {x_0 } |}) + g({| {c_0 } |})}]| {t - t'} |.
\end{aligned} \label{e2.23}
\end{equation}

 Case 2: $t$ and ${t'} $ are contained in $ 1 \le t \le T$.
 From \eqref{e2.15}, it follows that
\begin{equation}
\begin{aligned}
&Fb_m (t) - Fb_m ({t'}) \\
&= \int_0^1 {f({s,b_m (s) + x_0 ,\phi (s)})ds}
 + \int_1^t {f({s,b_m (s) + x_0 ,b_m ({s - 1}) + x_0 })ds}\\
&\quad  - \int_0^1 {f({s,b_m (s) + x_0 ,\phi (s)})ds}
  - \int_1^{t'} {f({s,b_m (s) + x_0 ,b_m ({s - 1}) + x_0 })ds} \\
& = \int_{t'}^t {f({s,b_m (s) + x_0 ,b_m ({s - 1}) + x_0 })ds} .
\end{aligned} \label{e2.24}
\end{equation}
 From this equality and the hypotheses, we have
\begin{equation}
\begin{aligned}
| {Fb_m (t) - Fb_m ({t'})} |
&\le | {\int_{t'}^t {| {f({s,b_m (s) + x_0 ,b_m ({s - 1}) + x_0 })} |ds} } |\\
&\le | {\int_{t'}^t {h(s)[{g({| {b_m (s)} | + | {x_0 } |})
 + g({| {b_m ({s - 1})} | + | {x_0 } |})}]ds} } | \\
&\le | {\int_{t'}^t {Ng({| {b_m (s)} | + | {x_0 } |})ds + I_3 } } |
\end{aligned}\label{e2.25}
\end{equation}
where
\begin{equation}
I_3  = \int_{t'}^t {Ng({| {b_m ({s - 1})} | + | {x_0 } |})ds.} \label{e2.26}
\end{equation}
By the change of variable, we have
\begin{equation}
I_3  = \int_{t' - 1}^{t - 1} {Ng({| {b_m (\sigma)} | + | {x_0 } |})d\sigma  
\le Ng({b + | {x_0 } |})({t - t'})} .
\label{e2.27}
\end{equation}
Using the above inequality in \eqref{e2.25}, we obtain
\begin{equation}
| {Fb_m (t) - Fb_m ({t'})} | \le 2Ng({b + | {x_0 } |})| {t - t'} |.
\label{e2.28}
\end{equation}

 Case 3: $t$ and ${t'} $ are respectively contained in $ [{0,1}) $ and
 $ [{1,T}]$. From \eqref{e2.14} and \eqref{e2.15}, it follows that
\begin{equation}
\begin{aligned}
&Fb_m (t) - Fb_m ({t'})\\
&= \int_0^t {f({s,b_m (s) + x_0 ,\phi (s)})ds} \\
&\quad  - \int_0^1 {f({s,b_m (s) + x_0 ,\phi (s)})ds}
 - \int_1^{t'} {f({s,b_m (s) + x_0 ,b_m ({s - 1}) + x_0 })ds} \\
& =  - \int_t^0 {f({s,b_m (s) + x_0 ,\phi (s)})ds
 - \int_0^1 {f({s,b_m (s) + x_0 ,\phi (s)})ds} } \\
&\quad  - \int_1^{t'} {f({s,b_m (s) + x_0 ,b_m ({s - 1}) + x_0 })ds} .
\end{aligned}\label{e2.29}
\end{equation}
 From \eqref{e2.29} and using the hypotheses, we have
\begin{equation} \begin{aligned}
&| {Fb_m (t) - Fb_m ({t'})} |\\
&\le \int_t^1 {| {f({s,b_m (s)  + x_0 ,\phi (s)})} |} ds \\
&\quad  + \int_1^{t'} {| {f({s,b_m (s) + x_0 ,b_m ({s - 1}) + x_0 })} |ds}\\
&\le \int_t^1 {h(s)[{g({| {b_m (s)} | + | {x_0 } |})
  + g({| {\phi (s)} |})}]} ds \\
&\quad + \int_1^{t'} {h(s)[{g({| {b_m (s)} | + | {x_0 } |})
 + g({| {b_m ({s - 1})} | + | {x_0 } |})}]} ds \\
&\le \int_t^1 {N[{g({b + | {x_0 } |}) + g({| {c_0 } |})}]} ds
 + \int_1^{t'} {N[{g({b + | {x_0 } |}) + g({b + | {x_0 } |})}]} ds \\
& \le M({t' - t}),
\end{aligned}\label{e2.30}
\end{equation}
where
\[
M = \max \{ {N[{g({b + | {x_0 } |}) + g({| {c_0 } |})}],
2Ng({b + | {x_0 } |})} \}.
\]
 From \eqref{e2.23}, \eqref{e2.28}, \eqref{e2.30}, we conclude that
$\{ {Fb_m } \} $ is equicontinuous. By Arzela-Ascoli theorem (see [4,7]),
the operator $F$ is completely continuous.

Moreover, the set $ U(F) = \{ {y \in B_0 :y = \lambda
Fy,\lambda  \in ({0,1})} \} $ is bounded, since for every
$y$ in $U(F)$ the function $ x(t) = y(t) + x_0 $ is a solution
of \eqref{e2.4}-\eqref{e1.2}, for which we have proved
$ \| x\| \le Q $ and hence $ \| y \| \le Q + | {x_0 }|$.
Now, an application of Lemma 1, the operator $F$ has a fixed
point in $ {B_0 } $. This means that \eqref{e1.1}-\eqref{e1.2} has
a solution. The proof is complete.
\end{proof}

\paragraph{Remark.}
We note that the advantage of our approach here is that,
it yields simutaneously the existence of solution of
\eqref{e1.1}-\eqref{e1.2} and maximal interval of existence.
In the special case, if we take $h(t)=1$ in \eqref{e2.2} and the
integral on the right hand side in \eqref{e2.2} is assumed to diverge,
then the solution of \eqref{e1.1}-\eqref{e1.2} exists for every
$T < \infty$; that is, on the entire interval $\mathbb{R}_+$. Our result
in Theorem 1 yields existence of solution of \eqref{e1.1}-\eqref{e1.2} on
$\mathbb{R}_+$, if the integral on the right hand side in \eqref{e2.2}
is divergent i.e., $\int_\alpha ^\infty  {\frac{{ds}}{{g(s)}} = \infty }$.
Thus Theorem 1 can be considered as a further extension of the well
known theorem on global existence of solution of ordinary differential
equation due to Wintner given in [16].

The next theorem deals with the Osgood type uniqueness result for the
solutions of \eqref{e1.1}-\eqref{e1.2}.

\begin{theorem} \label{thm2}
Consider  \eqref{e1.1} with
$f \in C({\mathbb{R}_+   \times \mathbb{R}^n  \times \mathbb{R}^n
,\mathbb{R}^n })$, under the initial conditions in \eqref{e1.2}.
Suppose that:
\begin{itemize}
\item[(i)] the function $f$ satisfies
\begin{equation}
| {f({t,x,y}) - f({t,\bar x,\bar y})} | \le h(t)[{g({| {x - \bar x} |})
+ g({| {y - \bar y} |})}],
\label{e2.31}
\end{equation}
where $h \in C({\mathbb{R}_+  ,\mathbb{R}_+  })$,
$g(u)$ is a continuous, nondecreasing function for $u \ge 0,g(0) = 0$;

\item[(ii)] let
\[
G(r) = \int_{r_0 }^r {\frac{{ds}}{{g(s)}},\quad ({0 < r_0  \le r})} ,
\]
with $G^{ - 1}$ being the inverse function of $G$ and assume that
$\lim_{r_0  \to  + 0}G(r) = \infty$, for any fixed $r$.
\end{itemize}
Then \eqref{e1.1}-\eqref{e1.2} has at most one solution on
${\mathbb{R}_+  }$.
\end{theorem}

\begin{proof} Let $x(t), y(t)$ be two solutions of equation \eqref{e1.1},
 under the initial conditions
\begin{equation}
x({t - 1}) = y({t - 1}) = \phi (t),\quad({0 \le t < 1}),\quad
x(0) =y(0) = x_0 , \label{e2.32}
\end{equation}
 and let $ u(t) = | {x(t) - y(t)}|$, $t \in \mathbb{R}_+  $.
We consider the following two cases.

Case 1: $0 \le t < 1$. From the hypotheses, we have
\begin{equation}
\begin{aligned}
u(t) &\le \int_0^t {| {f({s,x(s),\phi (s)}) - f({s,y(s),\phi (s)})} |} ds\\
& \le \int_0^t {h(s)g({| {x(s) - y(s)} |})} ds\\
& \le \varepsilon _1  + \int_0^t {h(s)g({u(s)})} ds,
\end{aligned}\label{e2.33}
\end{equation}
where $\varepsilon _1  > 0$ is sufficiently small constant. Now, an
application of Lemma 2 to \eqref{e2.33} yields
\begin{equation}
| {x(t) - y(t)} | \le G^{ - 1}
\big[{G({\varepsilon _1 }) + \int_0^t {h(s)ds} }\big].
\label{e2.34}
\end{equation}

Case 2: $1 \le t < \infty $. From the hypotheses, we have
\begin{equation}
\begin{aligned}
u(t) &\le \int_0^1 {| {f({s,x(s),\phi (s)}) - f({s,y(s),\phi (s)})} |} ds \\
&\quad + \int_1^t {| {f({s,x(s),x({s - 1})}) - f({s,y(s),y({s - 1})})} |} ds\\
&\le \int_0^1 {h(s)g({u(s)})} ds + \int_1^t {h(s)[{g({u(s)})
  + g({u({s - 1})})}]} ds \\
& = \int_0^1 {h(s)g({u(s)})} ds + \int_1^t {h(s)g({u(s)})} ds
  + \int_1^t {h(s)g({u({s - 1})})} ds \\
& = \int_0^t {h(s)g({u(s)})} ds + I_4 ,
\end{aligned}\label{e2.35}
\end{equation}
where
\begin{equation}
I_4  = \int_1^t {h(s)g({u({s - 1})})} ds.\label{e2.36}
\end{equation}
By the change of variable, we observe that
\begin{equation}
I_4  \le \int_0^t {h({s + 1})g({u(s)})} ds.\label{e2.37}
\end{equation}
Using \eqref{e2.37} in \eqref{e2.35}, we obtain
\begin{equation}
u(t) \le \int_0^t {[{h(s) + h({s + 1})}]g({u(s)})} ds
 \le \varepsilon _2  + \int_0^t {[{h(s) + h({s + 1})}]g({u(s)})} ds,
\label{e2.38}
\end{equation}
where $\varepsilon_2 > 0$ is sufficiently small constant.
Now, an application of Lemma 2 to \eqref{e2.38} yields
\begin{equation}
| {x(t) - y(t)} | \le G^{ - 1} [{G({\varepsilon _2 })
 + \int_0^t {[{h(s) + h({s + 1})}]ds} }]. \label{e2.39}
\end{equation}
To apply the estimations  in \eqref{e2.34}, \eqref{e2.39} to
the uniqueness problem, we use the notation $G({r,r_0 })$ instead of
$G(r)$ and impose  the assumption
$\lim_ {r_0  \to  + 0} G({r,r_0 }) =  + \infty$,
for fixed $r$, then we obtain $\lim_{r_0  \to  + 0}G^{ - 1} ({r,r_0 }) = 0$,
 see [15, p. 77]. From \eqref{e2.34}, \eqref{e2.39}, it follows that
$| {x(t) - y(t)} | \le 0$ for $t \in \mathbb{R}_+$ and hence
$x(t)=y(t)$ on $\mathbb{R}_+$. Thus, there is at most one solution
to \eqref{e1.1}-\eqref{e1.2} on $\mathbb{R}_+$.
\end{proof}

\paragraph{Remark} We note that the condition \eqref{e2.31} corresponds
to the Osgood type condition concerning the uniqueness of solutions
in the theory of differential equations (see [4, p. 35]).

\section{Perturbed equations}

In this section, we consider the difference-differential equation of
the form
\begin{equation}
x'(t) = A(t)x(t) + B(t)x({t - 1}) + f({t,x(t),x({t - 1})}),\label{e3.1}
\end{equation}
for $t \in J$ with the initial conditions \eqref{e1.2}.
The equation \eqref{e3.1} is treated as a perturbation of the linear system
\begin{equation}
x'(t) = A(t)x(t) + B(t)x({t - 1}),\label{e3.2}
\end{equation}
for $t \in J$ with the initial conditions \eqref{e1.2}, where
$A(t),B(t)$ are continuous functions on $J$ and $f,\phi$ are the functions
as in \eqref{e1.1}, \eqref{e1.2}. Following Sugiyama [14], let
$K(t,s)$ be a matrix solution of equations:
\begin{gather*}
\frac{\partial }{{\partial t}}K({t,s}) = A(t)K({t,s})
+ B(t)K({t - 1,s})\quad ({0 \le s < t - 1}), \\
\frac{\partial }{{\partial t}}K({t,s}) = A(t)K({t,s})\quad
({0 < t - 1 < s < t,0 \le s < t < 1}), \\
K({t,t}) = 1, \\
K({t,s}) = 0\quad ({ - 1 \le t < 0}).
\end{gather*}
The function $K(t,s)$ satisfying the above properties is called the
kernel function for equation \eqref{e3.2}. For more details concerning
the kernel function and its use in the study of various properties
of solutions of difference-differential equations with perturbed terms,
see Bellman and Cooke [1,2]. By means of the kernel function $K(t,s)$,
it follows that the solution of \eqref{e3.1} with \eqref{e1.2} considered
as a perturbation of \eqref{e3.2} with \eqref{e1.2} is represented by
(see [13, p.457])
\begin{equation}
x(t) = x_0 (t) + \int_0^t {K({t,s})f({s,x(s),x({s - 1})})ds,}
\label{e3.3}
\end{equation}
where $x_0 (t)$ is a unique solution of \eqref{e3.2} with \eqref{e1.2}.
It is easy to observe that (see [13]) the integral equations which are
equivalent to \eqref{e3.3} are
\begin{equation}
x(t) = x_0 (t) + \int_0^t {K({t,s})f({s,x(s),\phi (s)})ds,}\label{e3.4}
\end{equation}
for $0 \le t < 1$ and
\begin{equation}
x(t) = x_0 (t) + \int_0^1 {K({t,s})f({s,x(s),\phi (s)})ds}
+ \int_1^t {K({t,s})f({s,x(s),x({s - 1})})ds,} \label{e3.5}
\end{equation}
for $ 1 \le t \le T$.


The following theorems concerning the existence and uniqueness of solutions
of \eqref{e3.1}-\eqref{e1.2} hold.

\begin{theorem} \label{thm3}
Suppose that:
\begin{itemize}
\item[(i)] the function $f$ in \eqref{e3.1} satisfies  \eqref{e2.1},
\item[(ii)] the unique solution $x_0 (t)$ of \eqref{e3.2}-\eqref{e1.2}
is bounded, that is
\begin{equation}
| {x_0 (t)} | \le c,\label{e3.6}
\end{equation}
for $t \in J$, where $c $ is a positive constant,

\item[(iii)] the kernel function $K(t,s)$ for \eqref{e3.2} is bounded;
that is,
\begin{equation}
| {K({t,s})} | \le L,\label{e3.7}
\end{equation}
for $0 \le s \le t \le T$, where $L \ge 0$ is a constant, and for each
$t' \in J$,
\begin{equation}
\lim_ {t \to t'} \int_0^T {| {K({t,s}) - K({t',s})} |ds = 0,}
\label{e3.8}
\end{equation}
is satisfied for $ t \in J$.
\end{itemize}
Then \eqref{e3.1}-\eqref{e1.2} has a solution $x(t)$ defined on $J$
provided $T$ satisfies
\begin{equation}
\int_0^T {L[{h(s) + h({s + 1})}]ds
< \int_\beta ^\infty  {\frac{{ds}}{{g(s)}},} }
\label{e3.9}
\end{equation}
where
\begin{equation}
\beta  = c + \int_0^1 {Lh(s)g({| {\phi (s)} |})ds.}
\label{e3.10}
\end{equation}
\end{theorem}

\begin{theorem} \label{thm4}
Consider  \eqref{e3.1} with
$f \in C({\mathbb{R}_+ \times \mathbb{R}^n  \times \mathbb{R}^n,
\mathbb{R}^n }) $ and the conditions in \eqref{e1.2}, as
a perturbation of  \eqref{e3.2} for $ {t \in \mathbb{R}_+  } $
with \eqref{e1.2}.
Suppose that:
\begin{itemize}
\item[(i)] the condition (i) of Theorem 3 holds and the kernel
function $K(t,s)$ satisfies the condition \eqref{e3.7},

\item[(ii)] the conditions (i)-(ii) of Theorem 2 hold.
\end{itemize}
Then \eqref{e3.1}-\eqref{e1.2} has at most one solution on
${\mathbb{R}_+  }$.
\end{theorem}

The proofs of Theorems 3 and 4 can be completed by following the proofs
of Theorems 1 and 2 given above, with suitable modifications and closely
looking at the proofs of existence results given in [9,10].
We omit the details here.

\paragraph{Remark}
We note that our approach to the existence study of
\eqref{e1.1}-\eqref{e1.2} and \eqref{e3.1}-\eqref{e1.2} is different
from those used in [11-15] and we believe that the results given here
are of independent interest. For the study of numerical solution of
general Volterra integral equation with delay arguments of the
form \eqref{e3.3}, see [3].

\subsection*{Acknowledgements}
The author wishes to express his sincere thanks to the anonymous referee
and Professor J. G. Dix for helpful comments and suggestions.

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\end{document}